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# complex roots/fundamental theory of algebra help?

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Hi, this is probably a really easy question but I'm kind of stuck.

Find the roots of x^4=16, and therefor find all four roots of y^4=-16.

I know that x = 2, 2i, -2, and -2i but I can not for the life of me find what y is.

Thanks!

Guest Oct 29, 2017
edited by Guest  Oct 29, 2017
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#1
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y^4 = -16  All its roots are:

y = -2 (-1)^(1/4),    y = 2 (-1)^(1/4),    y = -2 (-1)^(3/4),    y = 2 (-1)^(3/4)

Guest Oct 29, 2017
#2
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Thanks. I’m sorry if this is too much to ask but could you explain why and how you got those answers?

Guest Oct 29, 2017
#3
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Solve for y:

y^4 = -16

Take 4^th roots of both sides.

Taking 4^th roots gives 2 (-1)^(1/4) times the 4^th roots of unity:

y = -2 (-1)^(1/4)    or y = 2 (-1)^(1/4)      or y = -2 (-1)^(3/4)       or y = 2 (-1)^(3/4)

Guest Oct 29, 2017

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